Well, this is supposed to be the first post, but I wanted to post the puzzle game as early as I could, so here we are.

First, what's the meaning of the title "{x|x∉x}"?
Well, quite literally, it's the set that shouldn't exist. 

What about the URL? Does "hyper-meta" have any meaning?
To be honest (I always am), I tried a few other names (variations of my real name) but apparently they already existed, so I chose this one and it worked. And I'm quite happy with this name, actually.
The name refers to a phenomenon that I noticed when I simulated the first order phase transition of Potts model on a hyperbolic plane.
First order phase transitions are everywhere in everyday life, e.g. water boiling/steam condensing, water freezing/ice melting, etc. It features latent heat - a discontinuity of energy, and discontinuities of other physical quantities. That is because the free energy has two minima separated by a barrier as a function of the order parameter. At the phase transition temperature, the two minima has the same height, but to go from one phase to the other, it must climb up the free energy barrier first.
The barrier grows as the size of the system grows. In macroscopic, at phase transition temperature, two phases of a matter can coexist, and there's interfacial free energy between them, which is proportional to the area of the interface. The two phase are called metastable phases. In the thermodynamic limit (i.e. the size of the system goes to infinity), the interfacial free energy also goes to infinity - meaning that, the probability that the entire system goes from one phase to another becomes zero. So, essentially, if we have a huge amount of water, at exactly zero temperature it will never completely become ice (there will be ice crystals of various sizes but they will disappear at some point).
The point is, this only happens at exactly the phase transition temperature, when the two minima have the same height. If it's a little different from that, say, the temperature is a little bit lower, then the solid phase has a lower free energy than the liquid phase. Now the situation is different.
Imagine an ice crystal has grown in the water. It still needs to climb up the free energy barrier, but because the interfacial free energy is proportional to the interface area while the free energy it reduces by converting water to ice is proportional to the volume. In Euclidean space of any dimension, for any given number K, there's always a size where the ratio of the volume to the surface is greater than K (with any appropriate units, of course). So, if the ice crystal is large enough that the (volume * free energy difference) is larger than (surface area * interface free energy), it's more likely for it to continue to grow (since the total free energy decreases).
So, in one word, in Euclidean space, at equilibrium, any matter only exist in one phase if the transition is first order, except at the exact phase transition temperature. If the temperature is a little different, no matter how little the difference is, they cannot coexist.

Things are very different in hyperbolic space. In hyperbolic space, the area of a sphere (or any shape that's not too bizarre) is proportional to the surface area.
That means, if we have an ice crystal in water in hyperbolic space, no matter how big it has grown, its volume/surface area is still a constant. Even if solid state has lower free energy - i.e. a global solid state would be in favor - it still cannot grow because the interfacial free energy grows faster, which makes it unlikely.
So, in hyperbolic space, there's a region of temperature where the phases can't change to each other - the phase of the matter is history dependent. In Euclidean space, this only happens at a single point - if it's above the phase transition temperature or below it, we know the phase; if it's exactly at the temperature, we can't be sure because both are possible. In hyperbolic space, there's a region of temperatures where the phase depends on the history: whatever the phase was, it will remain in that phase, making it hard to define the transition temperature.
This is something similar to but different from the metastable phases in Euclidean space, so I gave it the name "hyper-metastable phases" which only exist in hyperbolic space.

I'm not sure if there's any materials we can experiment on to analyze this behavior - maybe on some porous medium? Or in particle physics where the phase transition is first order and the space can be interpreted as hyperbolic through AdS/CFT correspondence. But nonetheless, it's an interesting phenomenon I found.